Can Chemistry be Reduced to Standard QM?

The mantra of modern physics is that Chemistry fundamentally is a branch of Quantum Physics based on Schrödinger's equation of Quantum Mechanics QM as Standard QM, which appears to be accepted in principle by a majority of chemists as reductionists, under the agreement that the reduction is yet to be realised, the big obstacle being that Schrödinger's equation is uncomputable.

There are some prominent chemists who have expressed clear anti-reductionist views pointing to inabilities of QM to describe fundamental aspects of chemistry:

• Hans Primas: emergence  
• Guy Woolley: molecular structure
• Norman Hunter: chemical bonds and reactivity 

and Roald Hoffman viewing chemistry as creative science beyond QM. It may be that in fact most chemists share such views and think of chemistry rather in terms of molecular structure in 3d with chemical bonds formed under chemical reactions, than as abstract StdQM.

RealQM is based on a new form of Schrödinger's equation, which has direct physical interpretation in terms of non-overlapping charge densities, closely connects to chemists conception of molecules as formed by atomic kernels surrounded by electrons carrying individuality by occupying different regions of space. RealQM thus opens to a form of Real Quantum Chemistry explored under this tag, where there is no gap between theory and chemistry reality since the Schrödinger equation of RealQM is readily computable. 

• Quantum chemistry is but a narrow subfield of theoretical chemistry and numerical quantum chemistry is nothing but a powerful tool. (Hans Primas in Chemistry, Quantum Mechanics and Reductionism).
• All that has happened (in the 20th century) is that the 19th century rupture between chemistry and physics has been patched over in framework of quantum chemistry. (Guy Woolley in Foundations of Molecular Quantum Electrodynamics) 

Secret of Covalent Chemical Bonding by RealQM

RealQM offers an easily understandable explanation on clear physical grounds of covalent chemical bonding, which has remained a mystery within Standard text book QM (see PS below).

To exhibit the physical mechanism forming a covalent bond, let us here as an exercise consider a 2d model of a X2 molecule formed by two X atoms, each with a +1 kernel surrounded by -1 electron charge density. We can think of X as 2d model of an H atom. The model is described in this post and is realised in this code. The real case in 3d is considered in this post.  The essential physics is Coulomb interaction between charges of same/different sign. The question is how a Coulomb chemical bond can be formed between two neutral atoms? 

We start running the code with the atoms separated (D=50) to get this output:

We see two essentially non-interacting 2d charge densities (red) together with mid line crosscuts of charge density (red), kernel potentials (blue) and electron potentials (light-blue, different sign) with a total energy of -0.9068. We see that the electron potentials (light-blue) are roughly half of the kernel potentials (blue), reflecting two-way interaction between electrons and one-way interaction between kernel and electron. The 2d energy here of -0.9068 is not the 3d energy -1.

We next run the code with smaller kernel distance (D=20) to get a smaller total energy  -1.05443:

We see that the left and right electron wave functions/charge densities now interact and meet at line orthogonal to the line between the kernels with continuity (and small/zero derivative). We understand that the decrease of total energy has two sources:

• Decrease of kinetic energy from the fact the electron charges meet with non-zero value at the free boundary, which does not require decay to zero with kinetic energy cost.
• Decrease of potential energy from the shaded region between the left kernel potential (blue) and left electron potential (light-blue) in interaction with the right electron charge density (red) and vice versa.

We decrease the distance further (D=15) to get essentially the same total energy as minimum, as the kernel repulsion takes over, running this code:

Altogether, RealQM offers an explanation on clear physical grounds in the form of Coulomb potentials of covalent chemical bonding. The essence is the Bernoullli free boundary between electron charges allowing concentration of electron charge between kernels without cost of kinetic energy. The secret is the decrease of potential energy from the above shaded area combined with large charge density at the free boundary. We understand that the bond is formed from a subtle interplay between electron charge distributions of substantial width with kernels of much smaller width. This is a new revelation by RealQM. Text book explanations struggle with “charge concentration”combined with “charge decentralisation” which appears contradictory.

PS chatGPT admits that there is no explanation in real physical terms of covalent bond.

 

Can an Unphysical Theory Explain Real Physics?

Recent posts pose the question to what degree standard Quantum Mechanics stdQM does explain (or can explain) the Periodic Table and chemical bonding. The standard answer is that for sure this is so, since stdQM gives a complete description of the world of atoms and molecules as the most successful theory of all of physics, and that this description of course includes the Periodic Table and chemical bonding. 

But this rosy picture can be questioned on several grounds, the most basic from the fact that there is no consensus after 100 years of brooding about the physical meaning of the  Schrödinger wave function $\Psi$ as the basic element of stdQM. But there is general understanding that $\Psi$ is unobservable as something without direct physical presence and in this sense is unphysical. 

The question then presents itself: Can an unphysical theory be used to explain real physics? 

The consensus is that stdQM correctly predicts outcomes of physical experiments, as far as tested. There is not a single experiment contradicting prediction by stdQM even if outcomes for the same experiment vary substantially.  Therefore stdQM is considered to be a theory that says something about the physics of atoms and molecules. Following a mantra of "shut up and calculate" it then appears as a black box capable of predicting outcomes of experiments, while the real physics is hidden to inspection. 

We can compare with a theory about the game of darts named Dart Mechanics DM stating that the spread of an inexperienced player will follow a normal distribution, which possibly can be verified by letting novice players play many games. The DM prediction of a normal distribution can thus be seen to agree with suitable experiments, but DM will say little about experienced players. More precisely, DM cannot explain anything about the real physics of the game of dart, since no such physics was put in. 

We are led to an idea that it may be very difficult to explain real physics using stdQM, since the physics of stdQM is hidden to inspection and so to explanation. In this perspective it is not strange that the efforts to explain the Periodic Table and chemical bonding have had mixed success, despite assertions that the success is total.

Here is what chatGPT has to say about stdQM vs real physics.

Discussion with Eric Scerri on the Periodic Table

Eric Scerri as world expert on the Periodic Table has offered interesting comments to this post asking if the Periodic Table is explained by standard Quantum Mechanics stdQM, in particular if the doubling of periods in the list 2, 8, 8, 18, 18, 32 and 32, is well explained. Scerri says that it has not yet been explained (after 100 years), but that it may be possible. 

The discussion touches the core of stdQM as the fermionic nature of electrons imbedded in anti-symmetric wave functions including a two-valued spin quantum number expressed in the Pauli Exclusion Principle PEP. 

Schrödinger formulated his equation for the Hydrogen atom with one electron in 1926, which gave meaning to quantum numbers introduced earlier and so was a formidable success. Schrödinger hinted at a direct formal extension to Helium with two electrons and so on for many electrons adding new spatial variables, but that model did not work because it made no distinction between bosons and fermions. So was Schrödinger's original model quickly complemented with a requirement that multi-dimensional wave functions must be anti-symmetric as fermions with satisfaction of PEP, which became stdQM. But Pauli was not happy with his PEP, even if it gave him the Nobel Prize in Physics, viewing it to be an ad hoc fix. 

RealQM gives a different generalisation of Schrödinger's so successful equation for the Hydrogen atom, where electrons have identity by occupying different regions of a common 3d space. There is here no need of a PEP nor of a 4th spin quantum number, in the original spirit of Schrödinger and Pauli. Electron configurations appear in RealQM as the result of a packing problem of non-overlapping charge densities. It appears that RealQM can add physics to the Periodic Table with doubling of periods as a natural outcome of electron packing under energy minimisation. 

RealQM Molecule Model 2D

To illustrate the basic feature of the new Schrödinger equation of Real Quantum Mechanics RealQM, let us  reduce from 3d to 2d, to get the following formulation for a molecule with $N$ electrons (like graphene):

Find the wave function $\Psi (x)$ with $x$ a 2d Euclidean coordinate, of the form 

• $\Psi (x) = \Psi_1(x) + \Psi_2 (x) +    + \Psi_N(x)$

where the $\Psi_n(x)$ for $n=1,2,...,N$ are one-electron wave functions with non-overlapping supports meeting a Bernoulli free boundary $\Gamma$, which minimizes the total energy 

• $E(\Psi ) = \frac{1}{2}\sum_{n=1}^N\int \vert\nabla\Psi_n\vert^2dx-\int P(x)\Psi^2dx$  (kinetic + potential energy)

with $P(x)$ a potential with contribution from electron and kernel charges, under the side condition 

• $\int\Psi_n^2 dx = 1$ for $n=1,2,...,N,$

and the Bernoulli free boundary condition:

• $\Psi (x)$ is continuous and 
• the normal derivative of $\Psi_n(x)$ vanishes on $\Gamma$ for $n=1,2,...,N.$  

This minimisation problem is solved with a gradient method realised as an explicit update consisting of three lines for iterative update of (i) wave function, (ii) level set function for $\Gamma$ and (iii) potential $P(x)$ realised in this code.

We consider a molecule consisting of two H atoms in green and one Beryllium atom with valence shell consisting of two electrons as two "half-shells" in red and blue. We start the iteration with the electrons concentrated into disks without overlap:

 

We find the following configuration after energy minimisation using this code to see the formation of a free boundary between electron wave functions:

We see that electron wave functions meet at a free boundary with continuity and zero normal derivative displayed in a horizontal cross-cut in yellow and vertical in magenta. We find a total energy of -12.34.

We compare with the atoms well separated with a total energy of -5.803 after energy minimisation.  

We understand that the reason the energy is lower for electrons in Bernoulli free boundary contact is that the charge density is concentrated between the kernels thus decreasing potential energy without increase of kinetic energy measured by $\nabla\Psi_n(x)$, since RealQM wave functions are not required to tend to zero like the global wave functions of stdQM.

Does Quantum Mechanics Explain Chemical Bonding?

This is a follow up of the recent post posing the question if quantum mechanics explains the Periodic Table connecting to the work by Eric Scerri as a world authority on the chemistry of the Periodic Table. In this article from 2023 A commentary on Robin Hendry’s views on molecular structure, emergence and chemical bonding Scerri starts out:

• There is a long-standing problem in the philosophy of chemistry arising from the fact that molecular structure cannot be strictly derived from quantum mechanics. 
• Two or more compounds which share a molecular formula, but which differ with respect to their structures, have identical Hamiltonian operators within the quantum mechanical formalism. 
• As a consequence, the properties of all such isomers yield precisely the same calculated quantities such as their energies, dipole moments etc. 
• The only means through which the difference between the isomers can be recovered is to build their structures into the quantum mechanical calculations, something that is carried out by the application of the Born-Oppenheimer approximation. 
•  I agree that chemists view bonding in a more realistic fashion and may consider bonds to be in some senses real, while physicists may consider bonding in more abstract energetic terms. 

We learn from Scerri that quantum mechanics does not distinguish between isomers like C2H5OH (ethanol)  CH3OCH3 (dimethyl ether) with vastly different chemistry from different bonding and so must be complemented by some form of expertise, which differs between chemists who seek explanation in some real terms and physicists satisfied with something purely abstract.

RealQM is a new form of quantum mechanics in terms of classical continuum mechanics as a system of non-overlapping charge densities with direct physical meaning in 3d space, which meets a chemist's demand of a reality. In RealQM the formation/bonding of a molecule can be simulated which can make a distinction between isomers without need of expertise. The window opened by RealQM is explored in posts on Real Quantum Chemistry. 

Non-Overlapping Wave Functions/Charge Densities

This is a follow up of the recent post on The Secret of Covalent Bonding with further computations comparing the non-overlapping wave functions/charge densities of RealQM meeting with (i) continuity and zero derivative, with a hypothetical case of instead (ii) zero density. 

To pin-point the essential aspect, we consider the following 1d model problem of an atom with N electrons: 

Find the function $\Psi (x)$ on the interval $[0,1]$ which minimises the energy $E=E_k + E_p$ with

• $E_p=\frac{1}{2}\int_0^1D\psi^2dx$             (kinetic energy)
• $E_k =-\int_0^1P(x)\psi (x)$           (potential energy)

over wave functions $\psi (x)$ with $D\psi =\frac{d\psi}{dx}$ the derivative, with total charge 

• $\int_0^1\psi^2dx = N$,

where $-P(x)$ is a given potential.

The Ansatz of RealQM is to seek $\Psi (x)$ on the form 

•  $\Psi (x) = \Psi_1(x) + \Psi_2(x) + .... + \Psi_N(x)$,

where the $\Psi_i(x)$ are one-electron wave functions with disjoint supports which meet on a Bernoulli free boundary with continuity and zero derivative. Running this code in a case with $N=4$ and $P(x)$ the potential from two kernels, we get the following result for case (i): 

We see in red 4 non-overlapping wave functions meeting at a free boundary marked by green with continuity and zero derivative. The total energy is -247 with kinetic energy 30. 

We run the same code but require wave-functions to vanish at meeting points and get for case (ii):

We see total energy larger total energy -231 with much larger kinetic energy 308. 

We understand that requiring wave functions to vanish as in (ii) when meeting, gives much larger kinetic energy than asking only for continuity and zero derivate as in (i), and that the net effect is larger total energy.

The purpose of this exercise is to exhibit the possibility opened by RealQM of electron concentration between kernels decreasing potential energy without balancing increase of kinetic energy, which appears as the holy grail of covalent bonding. 

In StdQM wave functions have global support, which may be overlapping, and so like the functions in (ii) above have to pay a price of added kinetic energy from charge concentration which is less favourable for binding than RealQM.

The finding of this post agrees with the real case considered in previous posts.

 

Does Quantum Mechanics Explain the Periodic Table?

Eric Scerri is a leading expert on the Periodic Table as the foundation of chemistry. This is what chatGPT says when asking the simple question to what extent the Periodic Table, formed before the emergence of quantum mechanics in the 1920s, has been explained by quantum mechanics during the now 100 years since then:

Eric Scerri, a philosopher and historian of chemistry, is known for his critical stance on the idea that quantum mechanics fully explains the periodic table. While he acknowledges the importance of quantum mechanics in understanding atomic structure and trends in the periodic table (such as electron configurations, periodicity, and chemical behavior), Scerri argues that the explanation is incomplete or limited if we rely solely on quantum mechanics.

Here are the key points of Scerri's position:

1. Quantum mechanics is useful but not fully explanatory: Scerri believes that while quantum mechanics (especially the Schrödinger equation and quantum numbers) helps rationalize some features of the periodic table, it does not provide a complete or entirely satisfactory explanation, especially for the actual order of elements.

2. Electron configurations are not derived solely from quantum mechanics: He points out that the Aufbau principle, which is used to predict electron configurations, is not a strict consequence of quantum theory but rather a semi-empirical rule. Some observed configurations (especially for heavier elements) deviate from the predicted ones.

3. Many-body problem: Scerri highlights the complexity of solving the many-body Schrödinger equation for atoms with more than one electron. Since exact solutions don't exist for such systems, approximations are required, which limits the explanatory power of quantum mechanics.

4. Philosophical perspective: He often argues from a philosophy of science perspective, suggesting that science often progresses through approximations, models, and partial explanations, rather than final, complete theories. Thus, he sees the periodic table as something emergent, not entirely reducible to quantum mechanics.

5. Chemistry's autonomy: Scerri also defends a degree of autonomy for chemistry, pushing back against the idea that chemistry is just applied physics. He sees the periodic system as a chemical construct that has its own foundations and logic, not fully subsumed under physics.

In summary, Scerri does not reject quantum mechanics, but he questions the common claim that it fully explains the periodic system. He calls for a more nuanced understanding of how different scientific disciplines interrelate, especially in the case of chemistry and physics.

We learn that Scerri as an expert in the field does not see that the Periodic Table can satisfactorily be explained by quantum mechanics and so calls for something more. But there is nothing more to grab. Quantum mechanics is supposed to be the best scientific model/explanation of the physics of atoms forming molecules forming chemistry. Quantum mechanics has fully replaced older atomic theories. If quantum mechanics cannot explain the Periodic Table, nothing can it would seem. 

The basic feature of the Periodic Table are the periods 2, 8, 18 and 32 of the form $2n^2$ with $n=1,2,3,4$, which appear as the number of atoms/elements in the rows of the table as 2, 8, 8, 18, 18, 32 and 32 thus with repetitions, as the number of electrons of the noble gases at the end of the row. What is the quantum mechanical explanation of these periods? 

The story is that this is because the eigenfunctions of the Schrödinger equation for the Hydrogen atom with one electron appear with a multiplicity of $n^2$ for $n=1,2,3,4$, which is doubled to fit with $2n^2$ by counting each electron twice. 

But the logic appears to be missing: Why expect the multiplicity of eigenfunctions of the Hydrogen atom with one electron to give the periodicity of atoms with many electrons? Why double the count of the number of electrons? Why is the periodicity repeated?

As concerns the sequence $2n^2$, there are many ways to this result. In RealQM it comes out as the solution to an electron packing problem partitioned into shells $n=1,2,3,...$ of increasing radius divided into half-shells filled with up to $n^2$ electrons allowing repetition of periods. This explanation carries the physical logic of packing of electrons of occupying different volumes depending on distance to the kernel.   

Stupid Swedish Foreign Policy

All political parties in Sweden from left to right have come to a historic consensus on foreign policy:

1. Trump is bad even if what he now does is to continue the politics of Biden: Genocide in Gaza, proxy war against Russia in Ukraine and bombing of the Houtis. 
2. Putin is bad and the only way to stop him from invading Sweden is to participate in the US/NATO proxy war against Russia in Ukraine with maximal military support.  

1. means that Trump will not come to help Sweden in case of Russian retaliation because of 2. Worst possible combination.

This is leading into catastrophe for the Swedish people, but opposition to this stupid policy is suppressed by very effective propaganda.

A better alternative is to come to a peaceful agreement with Putin by stoping sending Swedish weapons to Ukraine. This would open to rightful criticism of Trump without ruining Swedish society.

Form Classical to Modern to Post-Modern Physics

Classical physics in the form Newtonian mechanics emerged during the scientific revolution of the 18th and 19th centuries from an Enlightenment of rational logic/mathematics combined with observation of reality in a fundamental shift away from religion scholastics. The basic idea was rational mechanics as physics, which was (more or less) understandable and not only magical.  

In the late 19th century classical physics incorporated electro-magnetics made understandable through Maxwell's equations.  

But the modern physics of relativity theory and quantum mechanics emerging in the beginning of the 20th century signified a return to magical thinking. 

All the great physicists of the 20th century Einstein, Bohr, Schrödinger, Dirac, Feynman, Gell Mann, Weinberg admitted that quantum mechanics cannot be understood, while suggesting that this only adds to its beauty as the prime achievement of human intellect. 

To teachers of quantum mechanics this created a problem: How to teach a subject that is not understood by even the sharpest minds? The only way out for the average teacher was to pretend to understand and refer to the admissions of the top physicists of non understanding, as only a sort of teasing: Of course quantum mechanics is well understood and so can be presented in text books to students expected to come to understanding by diligent study. 

The computer is often presented as an off-spring of quantum mechanics thus showing its power, but the first realisations of the idea of a computer was made in rational mechanical terms as the Analytical Engine by Babbage with all its wheels and gears. A realisation in terms of quantum mechanics would be a quantum computer but the question is if there will ever be such a thing. 

Is it possible to imaging a form of post-modern physics with a return to the rationality of classical physics. My answer in all modesty is RealQM.